On the Essential Numerical Range

Abstract

The convexity. closure and compactness of the numerical range among other properties constitute a considerable literature in operator theory. The properties of the essential numerical range and how they are related to the familiar numerical range are studied. The study underpins the role in operator theory. An outline of the basic concepts and defined terms are provided. Siuiilarly, proofs for simple propositions and theo rems used ill the sequel cue made in Cluipter One. Cliapie: Two of this work is devoted to the properties of the numerical range. Properties, for instance, convexity. unitarv invariance and the projection property have been looked at. The counection between the spectrum of all operator and t he numerical range of rhe operator 111:1.s also been gi\'(~tl. The properties of the essential numerical range have been examined in Chapter- Thr-ee. The proofs of various properties studied, for instance, convexity have been outlined. In Chapter- FO'lLr-.the relationships between the numerical range <met the essential numerical range have been studied. The proofs of the theorems by J. Christophe elm I J. Lancaster have been shown. Finally. the roles of t.h« essential uurnerical range in operator theory have been discussed. Couclusious and reconnneudat ions for further research have also been givell.